Questions: Find the value of (x) and (YZ) if (Y) is between (X) and (Z). (XY=2x+1), (YZ doteq 6x), and (XZ=81) (X=square) (YZ=square)

Transcript text: Find the value of $x$ and $Y Z$ if $Y$ is between $X$ and $Z$. \[ \begin{array}{l} X Y=2 x+1, Y Z \doteq 6 x, \text { and } X Z=81 \\ X=\square \\ Y Z=\square \end{array} \]

Solution

Solution Steps

To find the value of $ x $ and $ YZ $, we can use the fact that $ Y $ is between $ X $ and $ Z $. This means that the sum of $ XY $ and $ YZ $ should equal $ XZ $. We can set up the equation $ XY + YZ = XZ $ and solve for $ x $. Once we have $ x $, we can substitute it back into the expression for $ YZ $ to find its value.

Solution Approach

Set up the equation $ XY + YZ = XZ $.
Substitute the given expressions: $ 2x + 1 + 6x = 81 $.
Solve for $ x $.
Substitute the value of $ x $ back into $ YZ = 6x $ to find $ YZ $.

Step 1: Set Up the Equation

Given the expressions: \[ XY = 2x + 1, \quad YZ = 6x, \quad \text{and} \quad XZ = 81 \] Since $ Y $ is between $ X $ and $ Z $, we have: \[ XY + YZ = XZ \]

Step 2: Substitute the Expressions

Substitute the given expressions into the equation: \[ 2x + 1 + 6x = 81 \]

Step 3: Simplify and Solve for $ x $

Combine like terms and solve for $ x $: \[ 8x + 1 = 81 \] \[ 8x = 80 \] \[ x = 10 \]

Step 4: Calculate $ YZ $

Substitute $ x = 10 $ back into the expression for $ YZ $: \[ YZ = 6x = 6 \cdot 10 = 60 \]